I'm reading Dwork's Boook An introduction to G-Functions and I'm stuck in some part of a proposition.
We say that $|-|$ is a valuation in the field $K$ (With values in $\mathbb{R}_{\geq 0}$) if
1) $|x|=0$ if and only if $x=0$
2) For all $x,y\in K$, $|xy|=|x||y|$, and
3) $|x+y|\leq\max\{|x|,|y|\}$.
We define the valuation group of $|-|$ by $G=\text{Im}|-|$. In the proof of the proposition which I'm stucked the key is to show that $G$ is dense in $\mathbb{R}$, but I cannot see why.
He says: '' Fix $\alpha\in K, \alpha\neq 0$ such that $|\alpha|\leq\varepsilon$'', where $\varepsilon$ is a fixed number greather than zero.
Can anyone give me any hint?
Thanks